2 The Einstein–Vlasov System

In this section we consider a self-gravitating collision-less gas in the framework of general relativity and we
present the Einstein–Vlasov system. It is most often the case in the mathematics literature that the speed of
light and the gravitational constant are normalized to one, but we keep these constants in the
formulas in this section since in some problems they do play an important role. However, in most
of the problems discussed in the forthcoming sections these constants will be normalized to
one.

Let be a four-dimensional manifold and let be a metric with Lorentz signature
so that is a spacetime. The metric is assumed to be time-orientable so that there is a distinction
between future and past directed vectors.

The possible values of the four-momentum of a particle with rest mass belong to the mass
shell , defined by

Hence, if , is the set of all future-directed time-like vectors with length , and if
it is the set of all future-directed null vectors. On we take and
(letters in the beginning of the alphabet always take values and letters in the middle
take ) as local coordinates, and is expressed in terms of and the metric in view of
Equation (25). Thus, the density function is a non-negative function on . Below we drop the
index on and simply write .

Since we are considering a collisionless gas, the particles follow the geodesics in spacetime. The
geodesics are projections onto spacetime of the curves in defined in local coordinates by

Here are the Christoffel symbols. Along a geodesic the density function is invariant so
that

which implies that

This is accordingly the Vlasov equation. We point out that sometimes the density function is considered as
a function on the entire tangent bundle rather than on the mass shell . The Vlasov
equation for then takes the form

This equation follows from (26) if we take the mass shell condition into account. Indeed,
by abuse of notation, we have

Here is considered as a function on in the left-hand side, and on in the right-hand side.
From the mass shell condition we derive

Inserting these relations into (26) we obtain (27). If we let , and divide the Vlasov
equation (26) by we obtain the most common form in the literature of the Vlasov equation

In a fixed spacetime the Vlasov equation (28) is a linear hyperbolic equation for and we can solve it by
solving the characteristic system,

In terms of initial data the solution of the Vlasov equation can be written as

In order to write down the Einstein–Vlasov system we need to know the energy-momentum tensor
in terms of and . We define

where, as usual, , and denotes the absolute value of the determinant of . We remark
that the measure

is the induced metric of the submanifold , and that is invariant under Lorentz
transformations of the tangent space, and it is often the case in the literature that is written
as

Let us now consider a collisionless gas consisting of particles with different rest masses
, described by density functions . Then the Vlasov equations for
the different density functions , together with the Einstein equations,

form the Einstein–Vlasov system for the collision-less gas. Here is the Ricci tensor, is the scalar
curvature and is the cosmological constant.

Henceforth, we always assume that there is only one species of particles in the gas and we write for
its energy momentum tensor. Moreover, in what follows, we normalize the rest mass of the particles,
the speed of light , and the gravitational constant , to one, if not otherwise explicitly stated that this
is not the case.

Let us now investigate the features of the energy momentum tensor for Vlasov matter. We define the
particle current density

Using normal coordinates based at a given point and assuming that is compactly supported, it is not
hard to see that is divergence-free, which is a necessary compatibility condition since the left-hand
side of (2) is divergence-free by the Bianchi identities. A computation in normal coordinates also shows that
is divergence-free, which expresses the fact that the number of particles is conserved. The definitions
of and immediately give us a number of inequalities. If is a future-directed time-like or
null vector then we have with equality if and only if at the given point. Hence, is
always future-directed time-like, if there are particles at that point. Moreover, if and are
future-directed time-like vectors then , which is the dominant energy condition.
This also implies that the weak energy condition holds. If is a space-like vector, then
. This is called the non-negative pressure condition, and it implies that the strong energy
condition holds as well. That the energy conditions hold for Vlasov matter is one reason that the
Vlasov equation defines a well-behaved matter model in general relativity. Another reason is
the well-posedness theorem by Choquet-Bruhat [55] for the Einstein–Vlasov system that we
state below. Before stating that theorem we first discuss the conditions imposed on the initial
data.

The initial data in the Cauchy problem for the Einstein–Vlasov system consist of a 3-dimensional
manifold , a Riemannian metric on , a symmetric tensor on , and a non-negative
scalar function on the tangent bundle of .

The relationship between a given initial data set on and the metric on the
spacetime manifold is, that there exists an embedding of into the spacetime such that the induced
metric and second fundamental form of coincide with the result of transporting with .
For the relation of the distribution functions and we have to note that is defined on the
mass shell. The initial condition imposed is that the restriction of to the part of the mass
shell over should be equal to , where sends each point of
the mass shell over to its orthogonal projection onto the tangent space to . An
initial data set for the Einstein–Vlasov system must satisfy the constraint equations, which read

Here and , where is the future directed unit normal vector to the
initial hypersurface, and is the orthogonal projection onto the tangent space to the
initial hypersurface. In terms of we can express and by ( satisfies , so it can
naturally be identified with a vector intrinsic to )

We can now state the local existence theorem by Choquet-Bruhat [55], for the Einstein–Vlasov
system.

Theorem 1 Letbe a 3-dimensional manifold,a smooth Riemannian metric on,asmooth symmetric tensor onanda smooth non-negative function of compact support on thetangent bundleof. Suppose that these objects satisfy the constraint equations (33, 34). Thenthere exists a smooth spacetime, a smooth distribution functionon the mass shell of thisspacetime, and a smooth embeddingofinto, which induces the given initial data onsuchthatandsatisfy the Einstein–Vlasov system andis a Cauchy surface. Moreover, givenany other spacetime, distribution functionand embeddingsatisfying these conditions,there exists a diffeomorphismfrom an open neighborhood ofinto an open neighborhood ofin, which satisfiesand carriesandtoand,respectively.

The above formulation is in the case of smooth initial data; for information on the regularity
needed on the initial data we refer to [55] and [118]. In this context we also mention that local
existence has been proven for the Yang–Mills–Vlasov system in [56], and that this problem for the
Einstein–Maxwell–Boltzmann system is treated in [30]. However, this result is not complete, as the
non-negativity of is left unanswered. Also, the hypotheses on the scattering kernel in this work leave
some room for further investigation. The local existence problem for physically reasonable assumptions on
the scattering kernel does not seem well understood in the context of the Einstein–Boltzmann system, and a
careful study of this problem would be desirable. The mathematical study of the Einstein–Boltzmann
system has been very sparse in the last few decades, although there has been some activity in recent years.
Since most questions on the global properties are completely open let us only very briefly mention some of
these works. Mucha [117] has improved the regularity assumptions on the initial data assumed
in [30]. Global existence for the homogeneous Einstein–Boltzmann system in Robertson–Walker
spacetimes is proven in [125], and a generalization to Bianchi type I symmetry is established
in [124].

In the following sections we present results on the global properties of solutions of the Einstein–Vlasov
system, which have been obtained during the last two decades.

Before ending this section we mention a few other sources for more background on the Einstein–Vlasov
system, cf. [156, 158, 73, 176].